Jacob Khurgin

Report
Away from High Loss in Plasmonics and
Towards
Low in
Loss
in Plasmonics
and Metamaterials
Battling Loss
Plasmonics
and
Metamaterials
Metamaterials
Jacob B Khurgin
Johns Hopkins University
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1
5 stages of dealing with the loss
1. Denial
2. Anger
3. Bargaining
4. Despair
5. Acceptance
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9000
80
8000
70
7000
60
6000
50
?
5000
40
4000
30
3000
20
2000
10
10,000 BC 1000
(discovery of Ag)
0
1990
Electron scattering time in Ag (fs)
Publications in Plasmonics and
Metamaterials (Web of Science )
Motivation
1995
2000
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year
2005
2010
2015
3
Scope
•Why are the metals necessary for sub-wavelength confinement?
•What are the surface plasmons and polaritons?
•Why subwavelength confinement in optical range always means high loss?
•Why reducing loss is so important? 3 Case studies
1. Who needs negative index?
2. How does loss impact plasmonic enhancement of the emission?
3. Plasmons and nonlinear optics –a winning combination?
•Why and how do the metals absorb and reflect?
•Can a metal be made lossless?
•Does it have to be a metal?
•Can metal loss be compensated by gain?
•Can one make a true sub-wavelength laser /spaser?
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Scope
•Why are the metals necessary for subwavelength confinement?
•What are the surface plasmons and polaritons?
•Why subwavelength confinement in optical range always means high loss?
•Why reducing loss is so important? 3 Case studies
1. Who needs negative index?
2. How does loss impact plasmonic enhancement of the emission?
3. Plasmons and nonlinear optics –a winning combination?
•Why and how do the metals absorb and reflect?
•Can a metal be made lossless?
•Does it have to be a metal?
•Can metal loss be compensated by gain?
•Can one make a true sub-wavelength laser /spaser?
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Why metal?
Metal’s claim to fame is its negative real part of the dielectric constant –Drude
Formula
Plasma frequency
 p2
 ( )  1 
Plasma frequency is defined as
 2  j
Scattering rate
Ne 2
 
 0m
2
p
Metal Plasma Frequencies and Scattering Rates
Plasma frequency
gold
silver
aluminum
Au
Ag
Al
2081 THz 8.5 eV
2182 THz 9.0 eV
3231 THz 13.2 eV
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Scattering rate
12.31013 s-1
3.2 1013 s-1
27.31013 s-1
6
Permittivity of Gold
Drude fit
experimental
Interband absorption
K. Busch et al., Phys. Rep. 444, 101 (2007)
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Material Q-factor
 Re( )
QM  
 Im( )
Realistically…QM~10-20 because of surface scattering
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What does negative  mean?
The electrons move to screen the electro-magnetic field – hence the field gets
expelled from the metal and the field inside becomes evanescent
E
ES
- Total Field=0
-
+
+
+
+
+
+
+
+
+
+
+
+
Metal
Refractive index –complex, mostly
imaginary
External field
Screening field
n()   ()  n()  j ()
Evanescent Field

E ~ E0e

~ E0e
  ( ) z
c
E
Surface Charge
x
j n ( ) z
c
z
Free Electrons
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Bulk Charge Oscillations
 ( )  1 
 p2
 2  j
In bulk metal when the frequency exceeds plasma frequency the electrons can no
longer follow the electric field and screen it – metal becomes transparent
 (p )  0
At plasma frequency the bulk metal supports longitudinal charge oscillations
EE
+
+
+
+
+
+
+
+
+
+
+
+
++++++++++++-
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Why do we need free carriers?
We want to concentrate optical field on subwavelength scale…but are prevented by the
diffraction limit
Lmin ~  / 2n
Comes from the uncertainty principle
px  / 2
k x  / 2
2 n
Lmin  1/ 2

Lmin ~  / 4 n
Easy to understand for real n…but
what about imaginary part?
To see how we can beat diffraction limit …let us re-derive it from the energy
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conservation considerations
11

a
Energy balance in a modea

t=0
t/2
E
H
n
k

E sin(kz)sin(t )
0
c H cos(kz) cos(t )
E H    E  nE
 H = 
k
0
2
Electric
t
2
2
E
Magnetic
H
E
“Potential” U  
e
U M  0

“Kinetic”
Energy
2
2
2
Energy
Energy oscillates
between potential
(electric) and kinetic
(magnetic)
2 n
UM
UE
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Lack of energy balance in a sub- mode
a

0
2 c
a

2n n
a

t=0
t=0
H
E
 
 
H cos  z  cos(t )
E sin  z  sin(t )
a 
 a  E
a
2a
2na nE
 H = 
H
 E  c E 
t

0
0 0
2
2
Electric
E 2 Magnetic
2
2
“Kinetic” U   H   2na   E   2na  U
“Potential” U e  



 E
M
0
2
2

2

Energy
Energy

If a<<0/2n there is almost
no magnetic field (quasistatic limit) UM<<UE –energy
is not conserved
UM
UE
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0


0

The energy will radiate
because it cannot all fit into
magnetic energy –this is
diffraction limit!
13
Free carriers restore balance in a sub- mode
a
t=0
0
2 c
a

2n n

t=0
+
+
+
E
 
2na nE
E sin  z  sin(t ) H 
a 
0 0
Electric
“Potential” U
e
Energy

a
2
E
2
v
J
2
True
Magnetic
 2na 
Kinetic”
“Kinetic” U M  
 UE
Energy of
Energy
 0 
electrons
At some resonant
frequency 0 the balance is
achieved
UE
UM
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UK

H  
H cos  z  cos(t )
a 
J cos(t )
Nmv 2 LK J 2
UK 
2
2
~  2 LK CU E
LK=Kinetic Inductance
(inertia of electrons)
If a<<0/2n there is almost
no magnetic field (quasistatic limit) UM<<UE – hence
energy oscillates between
electric energy and kinetic 14
energy of free carriers
Scope
•Why are the metals necessary for sub-wavelength confinement?
•What are the surface plasmons and polaritons?
•Why subwavelength confinement in optical range always means high loss?
•Why reducing loss is so important? 3 Case studies
1. Who needs negative index?
2. How does loss impact plasmonic enhancement of the emission?
3. Plasmons and nonlinear optics –a winning combination?
•Why and how do the metals absorb and reflect?
•Can a metal be made lossless?
•Does it have to be a metal?
•Can metal loss be compensated by gain?
•Can one make a true sub-wavelength laser /spaser?
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Surface charge oscillations
+- +- -+ +- +- +- -+ +- -+ +- +- +m<0 -+ -+ +- +- +- +- -+ -+ +- +- +- +-+ -+ -+ +- +- +- +- -+ -+ +- +- +-
Surface plasmon (SP)
d>0
 m (sp )   d (sp )  0
p
sp 
1 D
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Interface surface plasmon polaritons (SPP)
Surface plasmons couple with electromagnetic waves
Unfortunately…in a real metal
3
<0
 spp
dm

c d  m
2.8

SPP is a TM wave propagating along
the interface and evanescent in
both mediums, with dispersion
Energy (eV)
>0 weff

c
vg 


n
2.9
2.7
2.6
2.5
2.4
2.3
2.2 0
5
10
15
20
25
Wave number  (relative units)
Near the resonance interface SPP is characterized by a very small (sub-wavelength)
effective width weff, large effective index c/ and small group velocity
The losses are important!!!!!
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Localized SPP of a sub-wavelength nanoparticle
For sub-wavelength dimensions one can use electro-static
approximation and solve Laplace equation in stead of wave equation
  r l
 
a
 a r a
l 
Pl (cos  ) Emax,l 
l 1
ra
l 1
 a 


 r 

M
 0
+
D  0
r
ra

a
a

l  cos  Emax,l 
2
2
a



  ra

 r 
 0
2
Surface charge density
oscillations coupled with electric
field
2l  1
l 
 0 Emax,l Pl (cos  )
l 1
Most Important is the Dipole mode l=1
p
 m (1 )  2 D  0 1 
2 D  1
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Dipole SPP of a sub-wavelength nanoparticle
Electric Field
Emax
Dipole Moment
Radiative Decay:
p1  2 a3 0 Emax,1
3


2 2 a
 rad 


3 D  D 
Larger particle-larger antenna
+
The field is confined in a small effective volume !
Veff ,l
For dipole mode
Veff ,1 
2 a3
D
8 a3

(l  1)2  D
~ a3
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Larger particle – less field concentration
19
Scope
•Why are the metals necessary for sub-wavelength confinement?
•What are the surface plasmons and polaritons?
•Why does subwavelength confinement in optical
range always means high loss?
•Why reducing loss is so important? 3 Case studies
1. Who needs negative index?
2. How does loss impact plasmonic enhancement of the emission?
3. Plasmons and nonlinear optics –a winning combination?
•Why and how do the metals absorb and reflect?
•Can a metal be made lossless?
•Does it have to be a metal?
•Can metal loss be compensated by gain?
•Can one make a true sub-wavelength laser /spaser?
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The heavy price of having free carriers
t=/2
In the sub-wavelength metallic structures (in all three
dimensions !) half of the time almost all the energy is
stored in kinetic motion of electrons –where it is being lost
with the decay rate 2 of the order of 10 fs-1. Therefore the
rate of energy loss in truly sub-wavelength structure ,eff, is
always of the order of .
J
Case of propagating SPP
Here sub-wavelength
simply means very large
wave-vector
3
2.9
vg 
Energy (eV)
2.8

c


n
2.7
High Loss in
“true plasmon” region
2.6
2.5
2.4
2.3
Long Range SPP
2.2 0
5
10
15
20
Wave number  (relative units)
25
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Kinetic and Magnetic Inductances
Kinetic inductance is caused by the inertia
of electrons
a
v2
2 a I 2
I2
U K  Nm A2 a 
 LK
2
 0 p2 A 2
2
J
H
2 a
LK 
;
2
A p 0
2r
E
Split Ring Resonator
Magnetic Inductance
A-current area with skin effect
LM  0 a ln
U M LM

The ratio of energies U K LK
8a
r
goes to zero as the physical size is decreased
As particle gets progressively smaller more and more energy gets
stored in the form of kinetic energy of electrons
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Finding Effective Q of plasmonic mode
Qeff 
a
 LM  LK 

 eff
LK 
J
3
10
H
2a=/4
a=4r
2r
Split Ring Resonator -example
Q-factor
E
/8
2 Qm0
/16
10
/32
1
10
Border wavelength ~
0
10 -1
(where optics – displacement current and electronics
10 p
- conductivity current meet
 H  j r 0 E +  E
electronics
Au
optics
0
10
1
10
b
2
10
3
10
Wavelength (m)
U M 0 2a 2
~
  2 2a 2
UE
 0 r
When conductivity current dominates there is no wavelength dependence between E
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23
and H fields, hence H field can be large enough
Estimate of effective loss
Q of Drude metal structure actually gets
reduced with wavelength and only starts
getting better when < - THz region which is
not really optics –and even then LC circuit is
not a very high Q resonator!!!!!
2
10
1
10
optics
eff()/
0
10
-1
/32
10
/16
/8
-2
10
-3
2a=/4
electronics
10
-4
10
(b)
0
10
1
10
b
Wavelength (m)
2
10
That is why high Q resonators in electronics
include:
low loss inductances with high 
(energy stored in spin magnetization and not
kinetic energy of electrons)
Quartz crystals
Surface acoustic waves
/2 cavities ….
The true sub-wavelength region where kinetic inductance dominates (plasmonics) is inherently lossy and
various geometric tricks will not mitigate this loss significantly!
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Plasmonic and loss are inseparable
 
Confinement (1/a)
a /n
 
Conductivity current
Free carriers
ELECTRONICS
  p
a /n
 
a  /n
Reactive
current, free carriers
PLASMONICS
a /n
a  /n  
 
Reactive
(Displacement)
Conductivity current
current, bound
Bound carriers
carriers
Absorber
OPTICS
1/100nm
1/50m RBNI-14
Frequency (n/)
Reactive current of free
carriers can be confined on
sub wavelength scale but
then it cannot engender
magnetic field strong enough
to store the energy without
excessive loss
25
Scope
•Why are the metals necessary for sub-wavelength confinement?
•What are the surface plasmons and polaritons?
•Why subwavelength confinement in optical range always means high loss?
•Why reducing loss is so important? 3 Case studies
1. Who needs negative index?
2. How does loss impact plasmonic enhancement of the emission?
3. Plasmons and nonlinear optics –a winning combination?
•Why and how do the metals absorb and reflect?
•Can a metal be made lossless?
•Does it have to be a metal?
•Can metal loss be compensated by gain?
•Can one make a true sub-wavelength laser /spaser?
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What are plasmonic metal nanostructures good
for?
1. High field concentration near the metal-dielectric interface
enhance linear and nonlinear optical properties , such as electroluminescence, photoluminescence, Raman effect and so on.
2. High effective index and strong confinement of interface SPP can be used to develop
sub-wavelength passive and active photonic devices (nano-laser and “spaser”)
3. New “meta-materials” with unusual properties including negative refractive index
  0   0 n    0  =377

0

Bending with no reflection – perfect lensing
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Case study 1: Who needs negative
index?
Why do we need negative index material?
Presumably to get to the super-resolution……-Pendry’s superlens.
n
-n
n
Perfect near field focus
object
But we know that
the moment we
introduce finite loss
in the metal , the
resolving power of
this near field lens
deteriorates
drastically
image
150
  1
 p2
100
  j
2
50
Z(nm)

  1

2
p
2
0
-50
-100
-150
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0
X(nm))
10
20
28
30
What if we reduce (or eliminate) loss?
Create “artificial dielectric” with metal nanoparticles playing the role of atoms with
large polarizability
02
Q
 eff   d  f 2



f
d
0   2  j
Q(1   2 / 02 )  j
02  p2 / K Resonant frequency (K=3 for spheres)
f-filling fraction
Q  /
Near the resonance we can get fairly large values of
1/2
neff ~  eff
~ Q1/2
Then we can just make a conventional lens with high resolution /neff that would
give us a magnified image in the far field
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What does it mean?
n
-n n
4
object
image
metal - dielectric composite lens
3
Silver
2
superlens
1
6
3
1 -1
Loss(s )
With the current high metal loss
the Superlens does not really
perform
13
0.2 x 10
If the loss is reduced one can get higher
resolution and magnified image in far
field without negative n!
Reduced loss would change everything!
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Meta-Catch 22
A catch-22 is a paradoxical situation in which an
individual cannot avoid a problem because of
contradictory constraints or rules. Often these
situations are such that solving one part of a
problem only creates another problem, which
ultimately leads back to the original problem.
As long as metal loss stays what it is, negative index materials in
optical range are highly unfeasible
If one could reduce metal loss by an order of magnitude or more
the negative index materials and devices may become highly
unnecessary
Probably makes sense to see what we can do about loss
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Case study 2: how loss impacts
plasmonic enhancement of emission,
absorption, Raman….
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Plasmonic Enhancement of Radiative
Efficiency-why loss is critical
Surface-plasmon-enhanced light emitters based on InGaN quantum wells
KOICHI OKAMOTO, ISAMU NIKI1, ALEXANDER SHVARTSER, YUKIO NARUKAWA,TAKASHI MUKAI2 AND AXEL SCHERER
Nature Materials 3, 601 (2004)
>10 –fold enhancement
(some form the enhanced absorption)
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Origin of plasmonic enhancement
Radiative decay is proportional to the density of states

1
rad
Oscillator strength
e f

3D  5  103  f 3D m3 / ps 2  3 D
2 0m0
2
n 
4
 2 3  3
 c

3
2
About 1/THz.m3 @400nm

1
nrad
~
3D
phonon
~
4
Radiative time even for the
allowed transition~100ps
3
 ph
 ph
Nonradiative time can be much
shorter…because density of final states is
orders of magnitude higher
 rad
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1
 rad
 nrad
 1

1
 nrad   rad
 rad   nrad
34
“The bottleneck”
Density of states
Nonradiative
processes
 rad
radiation
1
 rad
 nrad
 1

1
 rad   nrad  nrad   rad
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Removing bottleneck
Nonradiative states
1
 nrad
1
 rad
3D photon states
 3D 
4
Purcell Factor:
 rad
1
FP ~
~
 r1  3 D
 3
1
 r11   rad
1 ~
1
 V 
High density “photon” states
How to increase photon density?
1. Reduce the volume V<<λ3 or
2. Reduce the frequencies range Q=/>>1
Original 1946 Paper –
28 lines, 5 equations
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36
“Dense photons”
Density of states can be changed only by coupling to other media
(polariton effect) or by coupling between counter-propagating photons
(microcavity, photonic crystals)
Propagating “dense photons” – polaritons, slow light, SPP

FP ~
gap
P
~ cvg1
3 D
0
vg=d/dk
c
k
Photon states
Polariton states
0 resonance can be atomic, Bragg…
Dense States are Slow States –
Do not couple well into true photons
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“Dense photons II”
Density of states can be changed only by coupling to other media
(polariton effect) or by coupling between counter-propagating photons
(microcavity, photonic crystals)
Localized “dense photons” – microcavity, localized SP
Veff
n-cavity resonance

cav  3 
FP ~
~
3D Veff 
Free photon states
Dense States are Confined
Do not couple well into true photons
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38
Re-emergence of the bottleneck
Nonradiative states
1
 nrad
1
 rad
1
 r11  FP rad
1 ~
1
 V 
3D photon states
 3D 
 rad
4
 3
Transfer from “dense” (high impedance)
to normal (low impedance) photons
“dense photons”
 nrad
Nonradiative decay of “dense photons”
More nonradiative states
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39
The bottleneck is alive and well…just shifted
down the line
Density
of states
Reservoir
of
dense photons
 rad
Nonradiative
processes
radiation
 nrad
radiation
Nonradiative
processes
Nonradiative decay
RBNI-14
40
So…what is the enhancement?
Nonradiative states
1
 nrad
1
 rad
3D photon states
Original radiative
efficiency
1
 rad 
 rad
Purcell enhancement
1
 r11  FP rad
“dense photons”
 nrad
Out-coupling efficiency
out
New radiative efficiency
rad ,1  out
1
FP rad
1
1
FP rad
  nrad
 rad
 nrad

1
1
 nrad   rad
 rad
  nrad
 rad

 nrad   rad
F1 
Overall enhancement
rad ,1
out
out


rad rad  (1  rad ) FP1 rad
Good emitter cannot be improved!!!!
RBNI-14
41
Trade offs
Use effective index  '   / k
p
p
D
as a parameter
104
3
2.9
Purcell Factor
10
2.8
Energy (eV)
2
Group Index=c/vg
Loss
100
Effective width
-2
10
1
2
3
4
Effective Index ’
More of a Photon
5
vg 

c


n
2.7
2.6
2.5
2.4
2.3
2.2 0
5
10
15
20
25
Wave number  (relative units)
More of a Plasmon
The tight confined
and slow SPP offer large enhancement
of radiation rate, but
they are lossy and difficult to couple
RBNI-14
outside
42
Main result for Propagating Interface SPP
F1
Substantial
improvement can
be achieved only
for very inefficient
emitters with
rad<1%
RBNI-14
43
Enhancement with localized SPP
Dipole Moment
p1  2 a3 0 Emax,1
Larger particle-better antenna
+
Effective Volume
 a3
Veff 
D
Smaller particle – better resonator
It is easy to see that the same particle cannot be a good antenna and a good resonator at
once!
Only the emitters whose efficiency is originally quite
poor can be significantly enhanced by SPP
Similarly, when it comes to absorption only very weak
absorbers can be enhanced by SPP
Maximum possible enhancement is on the order of
RBNI-14
Q
 rad

  nrad
44
Case study 3: Plasmons and nonlinear
optics
RBNI-14
45
Rationale:
Nonlinear optical interactions are quite interesting and important, yet are also very weak –
how can one improve it?
It is well known that if one used pulsed (mode-locked) laser and concentrate the same
average power into the high peak power with low duty cycle (d.c) efficiency of nonlinear
processes will increase
P
t
Can we do the same in the space domain and concentrate the same power into higher
local power density to increase the efficiency ?
Ag
Plasmonics as a ”silver bullet” for
nonlinear optics
RBNI-14
“Mode-locking in space?”
46
Plasmonic concentrators
-
+
-
-
+
+
+
-
+
+
+
+
+
+
 ( )  1 
 2  j
+
+
M. Stockman, P. Nordlander
-
+
-
+
+
+
- +
- +
Q~
r 
~ ~ 10  20
i 
2
 Elocal 
4
4
5
~
Q
~
10

10


 E 
2
 Elocal 
2
2
3

 ~ Q ~ 10  10
 E 
-
 p2
But:
Plasmonic concentration always brings loss
RBNI-14
47
Practical figure of merit
Switching
I pump
f
For nonlinear switching using XPM or SPM
fnl 
I signal

c
Ln2 I ~ 2
L

nmax ~ 
For wavelength conversion
~

c
2
Ln2 I pump ~
2

2
nmax ~ 1
Maximum interaction length is determined by absorption hence the ultimate figure of merit is
what is the a maximum phase shift achievable :
fmax ~ 2
And how close it is to 1…
L

RBNI-14
nmax
48
Enhancing nonlinear index
I pump
Ag
I sig
c(3)
E pump
Ag
Ag
Ag
Ag
Esig
f – volume filling factor
Ag
Ag
f
Ag
– mode overlap
RBNI-14
49
Assessing nonlinearity enhancement
c eff(3)
n2,eff
4
3
4
~
12
f

Q
~
10
~
12
f

Q
n2
c (3)
This sounds mighty good…..
What about absorption?
Maximum phase shift
 eff 
2 nd
fmax 
2
eff
Enhanced as much as few hundreds times
still, assuming
3 fQ
n2,eff I ~ 4 Q 3n2 I
This sounds really good…..except
n2  1013 cm2 / W (chalcogenide glass)
fnl ,max  1010 I
indicating that the input pump pump density must be in excess of 10GW/cm2 in order
to attain switching or efficient frequency conversion, meaning that while the length of
the device can get reduced manyfold, the switching power cannot and remains huge….
and the things only go further downhill from here on once it is realized that all of the
enhancement is achieved because the pump field is really concentrated by a factor of
of 1000 GW/cm2 –way past break down!50
RBNI-14
Q2 >100! Local “intensity” is now in excess
Saturation of nonlinearity
Even if one disregards optical damage nonlinear index change saturates
Borchers et al, “Saturation of the all-optical Kerr effect in solids”, Opt Lett 37, 1541 (2012)
So, what is the
real limit?
RBNI-14
51
A slightly better figure of merit
Re( psig ,nl )
Psig ,nl ~ 12 0 f  Q c
4
(3)
2
E pump Esig
 6 f Q 0 ( nlocal  Esig
2
Factor of Q2 makes perfect sense –because SPP
mode is a harmonic oscillator with a given Q –
changing local index shifts resonant frequency and
causes change in polarizability proportional to Q2

0 0

Assuming that maximum index change is limited by material properties to nlocal  nmax  0.01
the maximum phase shift is…
nl ,max ~
2
eff
3 f  Q 2 nmax
 Qnmax  0.01
The path to achieve either all-optical switching or efficient frequency conversion is
RBNI-14
52
less than obvious
Phase shift vs. distance
0
10
-1
1m2
13
n2  10 cm / W
P=1W
2
Phase Shift (rad)
10
f=10-3
-2
10
-3
10
-4
f=10-4
1m2
f=10-5
n2  1013 cm2 / W
10
-5
10
P=1W
f=10-6
-6
10 -5
10
-4
10
-3
10
-2
10
-1
10
0
10
Length (cm)
RBNI-14
53
Two ways to define figure of merit
Scientific approach
---
+ + +
+ --
What is the maximum attainable enhancement of
nonlinear susceptibility?
+
+
+
+
For c(2) enhancement is fQ3 ~102-103
For c(3) enhancement is fQ6 ~105-106
+
+ -- +
+
+ +
Engineering approach
For the nonlinear index type process
– what is the maximum phase shift
attainable at 10dB loss?
What would be the overall maximum attainable
result at ~one absorption length?
max~Qnmax~10-2<<
Not enough for all-optical switch
(or frequency conversion)
RBNI-14
54
Why such a conflicting result ?
Scientific approach: what matters is the relative improvement
Take very weak process with efficiency approaching 0….then if the end result is <<1
Result
a very large power
= 10
0
Engineering approach: what matters is the end result
Result = 0 × 10
Ag
a very large power
<< 1
Using metal nanoparticles for enhancement of second order
nonlinear processes may not be a “silver bullet” we are looking for.
Plasmonic enhancement is an excellent technique for study of nonlinear optical properties
(the higher order the better) and sensing using it, but it perhaps less stellar for any type
efficient switching, conversion, gating etc.
RBNI-14
55
Scope
•Why are the metals necessary for sub-wavelength confinement?
•What are the surface plasmons and polaritons?
•Why subwavelength confinement in optical range always means high loss?
•Why reducing loss is so important? 3 Case studies
1. Who needs negative index?
2. How does loss impact plasmonic enhancement of the emission?
3. Plasmons and nonlinear optics –a winning combination?
•Why and how do the metals absorb and reflect?
•Can a metal be made lossless?
•Does it have to be a metal?
•Can metal loss be compensated by gain?
•Can one make a true sub-wavelength laser /spaser?
RBNI-14
56
Ec
How do the metals absorb?
E
Conduction
Band 2
There have to be
Real transitions into the Real states
Intraband transitions –assisted by scattering
R

2
( finitial  f final
all
states
3
EF
2
1
( eE  k 
)
m
2
c
4
2
2
M scat
 (E final  Einitial  )
Involves the states within  from the Fermi level
At small frequencies only the states near
Conduction
Fermi level are involved , but at optical
Band 1
frequencies –all bets are off and everything
k depends on density of final states
3
Valence Band
Im( ) ~
 ( )
2 2

(

)
~
M scat  final
3

At even higher frequencies also
InterbandRBNI-14
transitions –band edge
57
Why are the metals lossier than
Semiconductors?
E
E
E
Semiconductor
Metal
 final ,met

EF
EF
 final ,sem
R ~  final
k
Valence Band


k
Valence Band
Density of final states is much higher in the metals
Of course plasma frequency is higher in metals, hence trade–off is inevitable, but
semiconductor is a viable alternative in theRBNI-14
THz regime
58
Why at optical frequencies loss does not
decrease dramatically with temperature?
E
To maintain energy conservation phonons must be involved
To maintain momentum conservation
Phonons must have wave vector
commensurate with the Fermi wave vector
ph
  ph (kF )
For low frequencies only absorption of
phonons is possible – this process is
   ph (kF ) proportional to the number of
phonons and thus highly temperature
~ 1  5THz
dependent
ph
EF
kF
k
Valence Band
RBNI-14
For high frequencies both absorption and
emission of phonons is possible – the
later
process has a spontaneous
component that does not depend on
phonon number and thus is temperature
independent
59
Temperature independent absorption with e-escattering
(b)
(a)
k2
E
E
k2
Hee
h
Two electrons are excited by a single photon
Hee
k4
h
EF
EF
k3
k1
k3
k1
Only Umklapp scattering is contributing
k
E
( k3  k1   ( k4  k2   g
k
(c)
E
(d)
 ee ( ) ~ (  / E F 
k2
h
k4
k2
h
k4
2
EF
EF
Hee
k3
Hee
k1
k3
k
k1
k
RBNI-14
The process becomes very
important at high
frequencies
60
How does the metal reflects-refracts?
E
Virtual transitions to real states
Conduction
Band 2
Intraband transitions-onto itself
Im( ) ~ Rv ~
( f initial  f initial  k
( eE 
)
2
k2 
Looks very Drude-like
Involves only the states
near the Fermi level
initial
states
Positive detuning

Ec

2
Negative detuning
Interband transitions
EF
Conduction
Band 1
k
Valence Band

 ( )
 ( )   ib ( ) 
j 3


2
p
2
 ib ~

initial
states
2
f
( Pif E /  
2
( E final  Einitial   
0
Involves all the states
below the Fermi level
p2  e2  f v2f / 3 0 Depends only on Fermi-level properties
Depends only on density of states
2 2
 (
)
~
M

scat final on all final states
RBNI-14
61
Scope
•Why are the metals necessary for sub-wavelength confinement?
•What are the surface plasmons and polaritons?
•Why subwavelength confinement in optical range always means high loss?
•Why reducing loss is so important? 3 Case studies
1. Who needs negative index?
2. How does loss impact plasmonic enhancement of the emission?
3. Plasmons and nonlinear optics –a winning combination?
•Why and how do the metals absorb and reflect?
•Can a metal be made lossless?
•Does it have to be a metal?
•Can metal loss be compensated by gain?
•Can one make a true sub-wavelength laser /spaser?
RBNI-14
62
Decoupling absorption and reflection
E
Conduction
Band 2
There is no final state for the
Absorption to take place
 p2
 ( )
 ( )   ib ( )  2  j 3


E12

Ec
p2  e2  f v2f / 3 0

 ( ) ~
EF
Conduction
Band 1
Ecv

Valence Band
2
2
M scat
 final  0
So intraband absorption is 0,
but what about interband?
Still 0 because there are no real states in the gap
RBNI-14
63
Lossless, yet metal?
E
Conduction
Band 2
There is no final state for the
absorption to take place
Ec    Ecv , E12
E12

Ec

But we still want negative 
 p2
 ( )   ib ( )  2

EF
Conduction
Band 1
Ecv
Thus we want narrow bands with wide gaps

Valence Band
2  p2  e2  f v2f / 3 0
We need large Fermi velocity (small effective
mass)
As always in Nature we see two
contradictory demands and thus should see
if some type of trade-off and compromise is
possible
RBNI-14
64
Consider b.c.c. lattice (Na)
Tight Binding Model
a
Vss
Coupling strength Vss
Ec
EF
Brillouin Zone –f.c.c.
RBNI-14
65
Lossless metal condition

1/2
16Vss    0.6e Vss / a 0 ib 
2
E12
p
Solution is possible but it take place at large interatomic distances that are bigger than in typical metals
(and also prone to Mott localization)
Ec
Lossy dielectric
Dielectric
Lossless metal
Lossy metal
0
Vss~exp(-a/a0)
a0/a
a
RBNI-14
66
Example:Na
0
-1
-2
3p
Energy (eV)
-3
-4
Ef
-5
E12
Ec
3s
-6
-7
-8
-9
-10
2
4
6
8
10
12
14
Lattice Constant (A)
Need Lattice constant of 8 Angstrom instead of 4.3 Angstrom
That is why metals absorb….
RBNI-14
67
Hypothetical 8 A lattice spacing Na
3
Interband
Absorption
2
1
0.6
Free
Carrier
Absorption
0.4
i
0.2
0
-1
-2
-3
12
1500
p
c
2000
2500
r
3000
Wavelength (nm)
RBNI-14
68
Big question: what do we do?
• We need to use stoichiometric arrangement of
metal atoms separated by the non-metal
atoms
• ITO –like materials are not solution – need
stoichiometry!
• Example: AlO –metal
• Maybe 2D structures?
O
Al
Al
Zn
AlZnO2
RBNI-14
69
69
Prospective materials
Na7MgF8
Na
F
Mg
Na
F
Mg
Mg
Na
9.7eV
Mg
F
Mg
RBNI-14
70
70
Scope
•Why are the metals necessary for sub-wavelength confinement?
•What are the surface plasmons and polaritons?
•Why subwavelength confinement in optical range always means high loss?
•Why reducing loss is so important? 3 Case studies
1. Who needs negative index?
2. How does loss impact plasmonic enhancement of the emission?
3. Plasmons and nonlinear optics –a winning combination?
•Why and how do the metals absorb and reflect?
•Can a metal be made lossless?
•Does it have to be a metal?
•Can metal loss be compensated by gain?
•Can one make a true sub-wavelength laser /spaser?
RBNI-14
71
Potential and Kinetic energy of the oscillator
Equation of motion
Amplitude
e / m 0
r0 ( )  2
0   2  j
d 2r
dr e
2



r


 E
0
dt 2
dt m
Dielectric constant
Ne2 / m 0
 ()  1  2
0   2  j
Potential Energy of electrons
UP 
Nm02 r0
4
2
Kinetic Energy of electrons
1
Ne2 / m 0
2 2
  0 E 0
2
2 2
4



( 0   2 2
UK 
Electric field Energy
Nm 2 r0
4
 Ne / m (    
1  (  
1
0
2
2
  UV  U P  U K
UE 
 0 E   0 E 1 
2
2
2
2
2
4 
4

(0       

Far from resonance   0  ( )  1 
2
0
2
1
UV   0 E 2
4
2
1
2 Ne / m 0
UP  0E
4
02
Ne2 / m 0
1
Ne2 / m 0 
2
U E   0 E 1 
  UV  U P
2
4

0


UP
1
Ne2 / m 0
2 2
 0E 
2
4
(02  2   2 2
“Pure” Electric field Energy
2
UV
2
02
UK  0
All of the electric energy is potential
(
UM
RBNI-14
0
0
72

In the dispersive region


2
2
2
2
 (  
1
1

Ne
/
m


Ne
/
m

2
2
0
0
0
Electric field Energy U E 

 0 E   0 E 1 

2
2
2
2
2
2
2
2
2
2
4 
4
 (0      
(1 0       

Potential Energy of electrons
2
“Non-dispersive (static)contribution” 4 (   0  E

1
Ne / m 0
1
U P   0 E 202

 0 E 2 
2
4
(02   2    2 2 4 
2

Ne / m 0
 Ne / m 0 

2
2 2
2
2 2
2 2



(
   2 2 
0
(0      
2
2
UP,S
Kinetic Energy of electrons
UP,D
1
Ne2 / m 0
2 2
UK  0E 
2
2 2
4



( 0    2 2
“Equal dispersive
(dynamic)contributions”
With dispersion –
additional energy
storage in dipole
oscillations –less need
for magnetic field

0
2
0

UV
UP,S
RBNI-14
UP,D
UM
UK
73
In the Reststrahlen region
(
Balance
(0
UV  UP,S  UP,D  UM  UK
0
0

Since
1
U P ,S  UV   E 2  0
4
U P,D  U K U M  0?
If we add the second dielectric with >0
UP  UV  UP,S  UP,D  UP,2  UM  UK
<0
2>0
No different from metal:
UP
In metal 0=0 hence there is no potential
energy
RBNI-14
UM
UK
74
Improvement?
What if we use dielectric, such as SiC in Reststrahlen region (10-12 microns)
2
2


LO
 TO
    1  2

2




j

TO



Re( )
TO   
QM 1 
QM 2 
2


Im( )



<0
LO
TO
SiC

>0

<0
SiO2
>0
Interface phonon polariton decay constant  is a few ps!!!! (not 10fs as in metal)
The width of SPP resonance eff~ gets reduced relative to the metal as it depends on QM1
It means that phonon polariton decays in time slower than plasmon polariton
But the propagation length depends on QM2< QM1 hence the propagation length is not much
longer(slow light effect)
1
Lprop  ( d / d    eff
Similarly, field enhancement is proportional to QM2< QM1 hence it is not as high as in metals
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U ~ ( /  E 2
Purcell effect is weak
Another way?
<0
Introduce highly dispersive
dielectric component: Rb atoms,
Quantum dots into the metaldielectric plasmonic structure…..
Ag
Ag

>0
SiO2
<0
SiO2
(
>0
0
0
Now significant amount of energy is in the form o fkinetic
energy of bound electrons – less energy goes into
conduction electrons oscillations-less scattering
UM
UM- Magnetic energy
UV- “pure E-field energy”
UP,S- “non-dispersive potential energy
of bound electrons
UP,D- “highly-dispersive potential
UP,S UP,D
energy of bound electrons
UV
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
UK,M-Kinetic Energy of
electrons in metal
UK,D
UM
UK,M
UK,D-Kinetic Energy of
bound electrons
76
“Slow-light loaded plasmon poariton”
Most Energy is contained in the oscillations of bound electrons in dielectric –
less in the oscillations of conduction electrons in the metal
The width of SPP resonance eff gets reduced relative to the metal
It means that SL loaded SPP decays in time slower than SPP with non-dispersive dielectric
But the propagation length does not change at all in SL loaded SPP (slow light effect)
Lprop  ( d / d    eff1
Similarly, field enhancement is SL-loaded SPP is no better than in normal SPP
U ~ ( /  E 2
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Can metal loss be compensated by
gain?
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Scope
•Why are the metals necessary for sub-wavelength confinement?
•What are the surface plasmons and polaritons?
•Why subwavelength confinement in optical range always means high loss?
•Why reducing loss is so important? 3 Case studies
1. Who needs negative index?
2. How does loss impact plasmonic enhancement of the emission?
3. Plasmons and nonlinear optics –a winning combination?
•Why and how do the metals absorb and reflect?
•Can a metal be made lossless?
•Does it have to be a metal?
•Can metal loss be compensated by gain?
•Can one make a true sub-wavelength laser /spaser?
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Two relevant questions
(1)Can one compensate the SP loss with gain?
<0
loss

>0
The rate of loss in the metal is on the scale of 1/(10 fs) – it
can only be compensated by the gain medium which has
high density of active atoms and allowed transitions semiconductor
gain
Semiconductor
Can one make a small “Nanolaser” or Spaser?
“spaser”
Metal
loss
Semiconductor gain
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Purcell Factor
Spontaneous decay is proportional to the density of photon states
8
1

~

=
In 3D space spon
3D
3
Metal

Veff
1
1
Localized SPP mode

~

~
LSP
LSP
For localized SP
Veff 
 spon  LSP
 3
3
Purcell Enhancement FP,LSP ~

~
~
Q ~ 102  104
 LSP 3D Veff  Veff
<0
>0

Metal
deff


vg =
∂ω
∂β
Dielectric
For propagating SPP
Propagating SPP mode
Purcell Enhancement
FP,LSP
 
d eff 
 spon  LSP  3
~

~
~ 101  103
 SPP 3D deff vg
But is it good or bad?
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1
 SPP
~  SPP ~
81
Can one compensate the plasmon loss with
gain?
Rate equations
<0
Ag Density of electron
loss

>0
gain
Semiconductor
current Purcell’s Factor
dnel
nel
I

 Fp
dt
eVeff
 rad
g  a(nel  ntr )
gain
differential gain
ntr ~ 1019 cm 3
Veff   3 / 20
Fp ~ 100;  rad ~ 10 10 s
1
J tr ~ eVeff1/3ntr FP rad
~ MA / cm 2
Transparency density
It is the current density and not the carrier density that matters.
Purcell Enhancement does play the role of a spoiler – not
surprising because in lasers we always want to reduce
spontaneous emission and not enhance it
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Compensation of loss in propagating SPP
E
z
M<0
2
z
E
S
2/qS
>0
Ez
-
Loss 
J
Gain g
J
Weff
x
n-doped
N Wa
kx
Rsp
Egap
p-doped
x=2/kx
+
Weff-90% energy effective width normalized to /n
neff
kx
/n


n / c
x
Effective modal index
Weff
Lx
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83
How does it look?
107
106
Transparency current
goes up as(1+neff5) (11/neff2)
Transparency current density
105
10
1
0
1019
eff (s-1)
10180
1014
neff
1013
0
2.2
1.8
0.1
“Real Plasmon”
Ntr(cm-3) Fp
1040
100
0.2
0.3
0.4
0.5
0.6
Purcell’s factor
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Long Range SP
Jtr(A/cm2)
Metal: Silver. To fit the SPP to the bandgap for wide range of wavelengths we need to use
“hypothetical” InxGaxNyAs1-y semiconductor
0.7
Weff
Transparency carrier density
0.1
0.2
0.3
0.4
0.5
0.6
Modal loss
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.7
0.1
500
Weff
Purcell’s factor
goes up as
(1+neff5)
Transparency
density is
reasonable
As expected, loss
goes up as
(1-1/neff2)
Weff
Effective modal index
1.4
10
Weff
0.2
0.3
0.4
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0.5
0.6
0.7
Weff
(nm)
84
Scope
•Why are the metals necessary for sub-wavelength confinement?
•What are the surface plasmons and polaritons?
•Why subwavelength confinement in optical range always means high loss?
•Why reducing loss is so important? 3 Case studies
1. Who needs negative index?
2. How does loss impact plasmonic enhancement of the emission?
3. Plasmons and nonlinear optics –a winning combination?
•Why and how do the metals absorb and reflect?
•Can a metal be made lossless?
•Does it have to be a metal?
•Can metal loss be compensated by gain?
•Can one make a true sub-wavelength laser /spaser?
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Enter The Spaser
LASER
Light
Amplification by
Stimulated
Emission of
Radiation
SPASER
Surface
Plasmon
Amplification by
Stimulated
Emission of
Radiation
Stockman, 2003
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N-doped AlGaAs
Is Spaser Unique?
++
+ +
+ +
d
Intrinsic GaAs
Barrier
AlAs
 ( d 
 gd 

Ag
barrier
10nm
Very small amount of radiation is coming out
for small SPASER
Energy is mostly contained
in the matter, not field
Barrier
AlAs
Fraction of energy contained in
free electron oscillations
P-doped AlGaAs
   na 2   d
f M  1  


   d   gd
 
 gd  1
fD 
 d   gd
   na 2 
1  
Total fraction of energy in the electron vibrations:
   d   gd  1
    
fe 
 d   gd
Fraction of energy contained in
bound electron oscillations in
dielectric (polariton)
For 40 nm GaAs Ag Spaser operating at 550nm
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fe  0.8
87
Is Spaser Unique?
+
+
+
+
R>99%
+
Fraction of energy contained in
free electron oscillations
Consider a small semiconductor laser with high
reflectivity mirrors Very small amount of radiation
is coming out of it
Fraction of energy contained in
 1
bound electron oscillations in f  gs
S
 s   gs
dielectric (polariton)
 gs  1
Total fraction of energy in the electron vibrations:
f 
fM  0
e
 s   gs
For GaAs laser operating at 880 nm –strong dispersion!
 ( s 
 (ns 
 gs 
  s  2ns
  s  2nng ~ 22


f  0.60
Energy is mostly contained
e
in the matter, not field
In any laser operating in a dispersive material not a photon but a polariton is emitted and
most of the energy is contained in electronic vibrations! The only difference is that in the
RBNI-14
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SPASER it is free electrons. So what’ s so special? Big loss!
Spaser and VCSEL
N-doped AlGaAs
Intrinsic GaAs
Barrier
AlAs
Ag
Barrier
AlAs
barrier
10nm
P-doped AlGaAs
VCSEL
MATTER
SPASER
Bound Electrons Energy
in semiconductor
36%
60%
Free Electrons Energy
in metal
30%
10%
45%
Magnetic Field Energy
Electric Field Energy
FIELD
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3%
16%
89
A very simple way to look at the spaser….
Holes flow
Spontaneous radiation
Rate equation for SP – bosonic particle
dN SP
 g eff ( N SP  1)   loss N SP
dt

The threshold is defined as
Metal
loss
N SP  1
Equal probability of stimulated and spontaneous emission
(linewidth is decreased by a factor of 2) geff   loss / 2
g
Semiconductor gain
 loss   metal   rad
Spontaneous SPP emission –energy transfer from
semiconductor to metal is at the rate of
loss
Electrons flow
Current is

I ~ e loss ~ 20 A
This best case result does not depend on shape, size( as long as it is truly sub-wavelength,
i.e. single mode) or on the gain material
Current density
6
J ~ I / πr 2 ~ 10RBNI-14
A / cm 2
90
What happens?
We want to reduce the current by reducing the volume
1
Itr ~ eVeff ntr FP rad
Unfortunately…Purcell factor is inversely proportional to the volume
The current scales down with the volume only until the dimensions
become sub-wavelength – after that – the same current has to go
into progressively ever smaller volume –not a very good idea!
The current is proportional to the number of modes (confined and
free space) into which the gain medium emits. The less is the
volume the less is the number of modes. Until you have just one
mode left. Then the volume stops being important.
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91
80 nm Single mode Au-InGaAs Spaser
“emitting” at 1320 nm
Active layer
P-contact
Au
N-contact
15
10
14
b
10
 loss
c
γeff -effective
linewidth
d
13
10
c
8
b
6
4
a
2
12
10
-7
10
-6
10
-5
10
It
-4
10
Current (A)
Input output characteristics shows
no threshold
0
0
It
100
200
Current (A)
Carrier density (gain) is
clamped near threshold –sign
of “spasing”
Linewidth narrowing indicates
“spasing threshold” of 28 A
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300
8
6
4
threshold
a
Number of SPP’s in the mode
 loss
12
10
d
“output”RSP
Carrier density (1018cm-3)
SPP emission rate (1/s)
and Linewidth (1/s)
10
2
0
0
2
4
Carrier density
6
8 Nc,th
(1018cm-3)
Near threshold half of SP’s are
coherent and half have random
phases – so it more like
“SPED” rather than “SPASER”
92
10
Lineshape evolution
1
-1
-2
-3
-4
-5
(a)
-6
0.7
0.8
0.9
Energy (eV)
0.5
Material Gain (1014 s-1)
0
-0.5
-1
-1.5
-2
-2.5
(b)
-3
0.7
1
0.8
1
Below threshold
Low pump
1.5
0.5
0
-0.5
-1
-1.5
(c)
-2.5
0.7
0.8
0.9
Energy (eV)
At threshold
1
Material Gain (1014 s-1)
Modal gain spectral density (a.u)
1.5
1
-2
0.9
Energy (eV)
Modal gain spectral density (a.u)
Material Gain (1014 s-1)
0
Modal gain spectral density (a.u)
Modal gain spectral density (a.u)
1
0.5
0
-0.5
-1
-1.5
(d)
-2
0.7
1
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0.8
0.9
Energy (eV)
1
Way above threshold
93
The threshold current depends on
Scattering Rate in Metal
The threshold current does not depend on
•Shape of the Mode
•Size of the mode as long as it is sub-wavelength in all 3 Dimensions
•Confinement factor of the mode
•Gain material! (as long as it works in a “normal laser”)
•Temperature (weakly)
•Wavelength (red, orange, IR, polka-dot) as long it is less than ~20m
•Longitude, Latitude, Altitude
•Attitude of the scientist
•Amount of money spent
Ithr = eγeff nsp ~ 20μA
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Do we need a nanolaser?
• True sub-diffraction structure always require metal and thus
inherently very lossy
• Therefore, threshold of the sub-diffraction laser will always be
very high (20 A in a tiny volume)
• There will be only a few photons (plasmons) in a mode – no
coherence (linewidth of THz)
• But the device will be very fast (THz)
What if we just use spontaneous emission –
SPED (Surface Plasmon Emitting Diode)?
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95
Characteristics of SPED
•
•
•
•
Small volume –high density of integration
Low power consumption
Easier to cool
High speed due to Purcell Effect
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Linewidth (Hz)
Intracavity
Power (W)
VCSEL, SPASER and SPED
10-2
VCSEL
10-3 (a)
SPASER
-4
10
LED
SPED
10-5
-6
10
10-7
10-8
102
103
104
105
106
Pump Current density (A/cm2)
1014
1012
(b)
1010
10
1013
1012
108
107
108
107
108
SPASER
LED
8
VCSEL
106102
Maximum
frequency (Hz )
SPED
107
103
104
105
106
Pump Current density (A/cm2)
(c)
SPASER
VCSEL
11
10
1010
9
10
108102
LED
103
SPED
104
105
106
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Pump Current density (A/cm2)
97
Conclusions:
•Loss affects and limits everything that is purportedly good about plasmonics and
metamaterials Reducing loss would change the paradigm.
•To get sub-wavelength concentration of light in all three dimensions one does need
material with <0 (usually a metal)
•In optical and near IR region it always leads to losses commensurate with rate of decay in
metal -1/10 fs, but far IR and THz the situation is less dire –perhaps mid-IR I swhere the
action should be
•Primarily because of loss only very weak optical processes can be enhanced by plasmonic
means (SERS) – any optical device that requires high efficiency cannot be improved by
plasmonic means. Hence sensing is the most promising niche.
•Loss in metals in optical range is fundamentally different from loss (resistance) at low
frequencies.
•Loss and negative dielectric constant can be decoupled (in the future)
•Using phonon polaritons and other resonant schemes reduces loss but has drawbacks of
its own
•Compensating loss with gain is tricky and probably unrealistic
•Incoherent sub-wavelength sources are just as good as lsub-waveelngth lasers and might
have a future.
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98
Additional reading
J. B. Khurgin, G. Sun “In search of elusive lossless metal, Appl. Phys. Lett, 96, 181102 (2010)
J. B. Khurgin, G. Sun, Enhancement of optical properties of nanoscaled objects by metal nanoparticles”, J. Opt. Soc. B, 26,
83 (2009)
J. B. Khurgin, G. Sun, Impact of high-order surface plasmon modes of metal nanoparticles on enhancement of optical
emission, Appl. Phys. Lett., 95, 171103 (2009)
J. B. Khurgin, G. Sun, “Enhancement of light absorption in quantum well by surface plasmon polariton”, Appl. Phys. Lett,
94, 191106 (2009)
J. B. Khurgin, G. Sun, R. A. Soref, “Plasmonic enhancement of photoluminescence by metal nanoparticles”, Appl. Phys.
Lett. 94, 101103 (2009);
J. B. Khurgin, G. Sun, R. A. Soref, “Practical limits of absorption enhancement near metal nanoparticles”, Appl. Phys. Lett.
94, 071103 (2009)
G. Sun, J.B. Khurgin, R. A. Soref, “Plasmonic light-emission enhancement with isolated metal nanoparticles and their
coupled arrays “, J. Opt. Soc. Am. B 25, 1748 (2008)
J.B. Khurgin, G. Sun, R. A. Soref “Electroluminescence efficiency enhancement using metal nanoparticles”, Appl. Phys.
Lett., 93 021120 (2008)
Khurgin JB, Sun G, Soref RA ”Enhancement of luminescence efficiency using surface plasmon polaritons: figures of merit” J.
Opt. Soc. Am. B 24: 1968-1980 (2007)
Khurgin JB ”Surface plasmon-assisted laser cooling of solids”, Phys. Rev. Lett 98, Art. No. 177401 (2007)
Sun G, Khurgin JB, Soref RA ”Practicable enhancement of spontaneous emission using surface plasmons” Appl. Phys. Lett.,
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90 Art. No. 111107 (2007)
99
J. B. Khurgin, A. Boltasseva, “Reflecting upon the losses in plasmonics and metamaterials”, MRS Bulletin , 37, 768-779 (2012)
J. B. Khurgin , G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium”,
Appl. Phys. Lett, 100, 011105 (2012)
J.B. Khurgin and G. Sun: “How small can “ Nano ” be in a “ Nanolaser ”?”, Nanophotonics, 1, 3-8 (2012)
J. B. Khurgin, G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence”, Optics
Express 20 15309-15325 (2012)
G. Sun, J. B. Khurgin, “Origin of giant difference between fluorescence, resonance, and nonresonance Raman scattering
enhancement by surface plasmons”, Phys. Rev. A 85, 063410 (2012
JG. Sun, J. B. Khurgin, and D. P. Tsai, “Comparative analysis of photoluminescence and Raman enhancement by metal
nanoparticles”, Opt. Letters, 37, 1583-1585 (2012)
G. Sun, J. B. Khurgin, A. Bratkovsky “Coupled-mode theory of field enhancement in complex metal nanostructures “, Phys. Rev, B
84, 045415 (2011)
J. B. Khurgin, G. Sun, Scaling of losses with size and wavelength in Nanoplasmonics” Appl. Phys. Lett, 99, 211106 (2011)
B. Zhang, J. B. Khurgin, “Eigen mode approach to the sub-wavelength imaging with surface plasmon polaritons” Appl. Phys. Lett,
98, 263102 (2011)
G. Sun, J. B. Khurgin, Optimization of the nanolens consisting of coupled metal nanoparticles: An analytical approach”, Appl.
Phys. Lett, 98, 153115 (2011)
G. Sun, J. B. Khurgin, “Plasmon Enhancement of Luminescence by Metal Nanoparticles : IEEE J. of Selected Topics In Quantum
Electronics , 17 ,110 (2011)
J. B. Khurgin, G. Sun , “Theory of optical emission enhancement by coupled metal nanoparticles: An analytical approach”, Appl.
Phys. Lett 98, 113116 (2011)
G. Sun, J. B. Khurgin, Comparative study of field enhancement between isolated and coupled metal nanoparticles: An analytical
approach”, Appl. Phys. Lett. 97, 263110 (2010)
RBNI-14
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Extra
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101
Higher order modes
High order modes tend to cling to the surface they are non-radiative
l=2
l=5
l=10
Veff ,l
8 a3

(l  1)2  D
Small volume– high field
concentration
E
  r l 1
  r a
a
E1 ~ e j1t Pl (cos  ) Emax,1 
l 2
 a 
ra
 r 

r
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a
102
Impedance mismatch
In electronics in order to go from the high impedance
circuit element (output stage of power amplifier-10K ) to the
low impedance medium (coaxial cable -75, strip line) one uses
Impedance transformer e.g. source follower
VDD
vin
IREF
vout
-VSS
Now we want “dense photons”
to serve as a buffer between the
emitting medium and the
propagating photons
Can this buffer be lossless?
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103
Impedance mismatch
In electronics in order to go from the high impedance
circuit element (output stage of power amplifier-10K ) to the
low impedance medium (coaxial cable -75, strip line) one uses
Impedance transformer e.g. source follower
VDD
vin
IREF
vout
-VSS
Now we want “dense photons”
to serve as a buffer between the
emitting medium and the
propagating photons
Can this buffer be lossless?
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104
Threshold of a laser with just one mode
Bjork, G., Karlsson, A. & Yamamoto, Y. Definition of a laser threshold.
Phys. Rev. A 50, 16751680 (1994).
We propose that the threshold of a laser is more appropriately described by the pump
power (or current) needed to bring the mean cavity photon number to unity, rather than
the conventional“ definition" that it is the pump power at which the optical gain equals the
cavity loss. In general the two definitions agree to within a factor of 2, but in a class of
microcavity lasers with high spontaneous emission coupling efficiency and high absorption
loss, the de6nitions may differ by several orders of magnitude.
J. B. Khurgin, G. Sun, “Injection pumped single mode surface plasmon
generators: threshold, linewidth, and coherence”, Optics Express 20 15309-15325
(2012)
..it corresponds to the number of SPs at threshold roughly Nsp,t~1 i.e., on average, just about one
SP in the mode, which is the threshold definition according to [36].”
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105
Single mode spaser as a “threshold-less” laser
Rate equations
Effective linewidth eff   loss  geff
Excess noise factor ~1
dN SP
 ( g eff   loss  N SP  g eff nspon
dt
dN c
 e 1 I ex  g eff N SP  g eff nsp  rspon  rA
Carriers
dt
SP
Auger recombination
(proportional to volume)
Excitation current
Spontaneous emission into other modes
If we neglect emission in other modes and nonradiative recombination…
Rate of SP generation is
RSP  NSP /  loss  I ex / e
N SP  I ex / e loss
No threshold!
Since due to small volume SP emission into the single mode is the dominant recombination
mechanism, all the energy goes into SP’s, but are they coherent?
We choose the critical effective linewidth narrowing
It corresponds to
geff ,c   loss / 2 and
Critical (threshold?) current
 eff ,c   loss  geff ,c   loss / 2
N SP,c  nsp  1 one SP in the mode
I cr  e eff nsp ~ 20 A
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